15 IoT Mathematics: Linear Algebra and Probability

15.8.1 C.1 Vectors

Notation: $\vec{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ or simply $[x, y, z]$

IoT Applications:

Vector Operations:

Magnitude (Length): How “big” is the vector? $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$

Example: Accelerometer reads $[0.5, 0.3, 0.8]g$ $|\vec{a}| = \sqrt{0.5^2 + 0.3^2 + 0.8^2} = \sqrt{0.98} \approx 0.99g$

This tells us the total acceleration magnitude (useful for fall detection).

15.8.2 C.2 Matrices

Notation: $M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

IoT Applications:

1. Coordinate Transformation:

Rotating sensor data to align with a reference frame:

2. Kalman Filter State:

State vector might be $[position, velocity, acceleration]$: $\vec{x} = \begin{bmatrix} p \\ v \\ a \end{bmatrix}$

State transition predicts next state: $\vec{x}_{k+1} = F \cdot \vec{x}_k$

3. Covariance Matrix:

Describes uncertainty in measurements: $P = \begin{bmatrix} \sigma_x^2 & \sigma_{xy} \\ \sigma_{xy} & \sigma_y^2 \end{bmatrix}$

Diagonal elements: Variance (uncertainty) in each variable Off-diagonal: How variables correlate (change together)

15.8.3 C.3 Matrix Multiplication

Example: Transforming a 3D point $\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}$

This scales the vector by 2 in all directions.

15.9.1 D.1 Basic Probability

Key Concepts:

Practical Example - Sensor Reliability:

If a sensor has 99.9% uptime:

• P(working) = 0.999
• P(failure) = 0.001

For 100 sensors, expected failures = 100 x 0.001 = 0.1 per unit time

15.9.2 D.2 Gaussian (Normal) Distribution

$p(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Properties:

• 68% of values within mu +/- 1 sigma
• 95% of values within mu +/- 2 sigma
• 99.7% of values within mu +/- 3 sigma

IoT Application - Sensor Noise:

A temperature sensor with:

• Mean reading: mu = 25.0 C
• Noise: sigma = 0.5 C

95% of readings will be between 24.0 C and 26.0 C.

15.9.3 D.3 Sensor Fusion: Combining Measurements

When you have two noisy measurements of the same thing, the optimal combination is:

$x_{combined} = \frac{\sigma_2^2 \cdot x_1 + \sigma_1^2 \cdot x_2}{\sigma_1^2 + \sigma_2^2}$

Intuition: Weight each measurement inversely by its uncertainty. Trust the more precise sensor more.

Example:

• GPS says position is 100m (sigma = 10m)
• Accelerometer integration says 95m (sigma = 5m)

$x_{combined} = \frac{25 \times 100 + 100 \times 95}{100 + 25} = \frac{2500 + 9500}{125} = 96m$

The combined estimate is closer to the accelerometer (more precise).

15.10.1 E.1 The Modulo Operation

Modulo is the arithmetic mechanism behind the wraparound behavior used throughout this section. Begin with small remainders to verify the operation, then carry the same rule into modular exponentiation. The examples establish the computation; later cryptographic sections add the essential requirements for large parameters, randomness, authentication, and safe key handling.

Notation: $a \mod n$ = remainder when a is divided by n

Examples:

• $7 \mod 5 = 2$ (7 = 1x5 + 2)
• $25 \mod 7 = 4$ (25 = 3x7 + 4)
• $12 \mod 12 = 0$ (clock wraps at midnight)

15.10.2 E.2 Why Modular Arithmetic for Cryptography?

Key Property: Easy to compute forward, hard to reverse.

Example - Discrete Logarithm:

Given: $g = 5$, $p = 23$

Forward (easy): Calculate $5^7 \mod 23$ $5^7 = 78125 \mod 23 = 17$

Reverse (hard): Given 17, find x where $5^x \mod 23 = 17$

This “trapdoor” property enables public-key cryptography.

15.10.3 E.3 Diffie-Hellman Key Exchange

Two IoT devices can agree on a shared secret over an insecure channel. First read the message sequence in Figure 15.1 from public parameters through exchanged public values to the independently derived shared result.

Follow Figure 15.1 from the public parameters through the exchange. Each device keeps its private exponent, publishes only a modular result, receives the other public value, and reaches the same shared value by combining it with its own secret. Figure 15.2 now restates the one-way property visually: follow public yellow into each private mixture, across the public exchange, and into the matching final colour.

Read Figure 15.2 from public yellow through the two private mixtures, across the public exchange, and into the matching final brown. The eavesdropper sees the public starting colour and both exchanged mixtures but cannot reverse them to recover either private colour. Likewise, an observer may see public values such as 8 and 19, while recovering the private exponents requires solving the discrete logarithm problem for properly chosen large parameters. The analogy explains one-way computation, not authentication; a real exchange still needs authenticated peers and safe key handling.

15.11.1 F.1 What Are Complex Numbers?

Form: $z = a + bi$

• $a$ = real part
• $b$ = imaginary part

Alternative (Polar) Form: $z = r \cdot e^{i\theta}$

• $r$ = magnitude (amplitude)
• $\theta$ = phase angle

15.11.2 F.2 Why Complex Numbers in IoT?

1. Representing Sinusoidal Signals:

A sensor reading: $v(t) = A \cos(\omega t + \phi)$

Using Euler’s formula: $e^{i\theta} = \cos\theta + i\sin\theta$

We can write: $v(t) = \text{Re}[A \cdot e^{i(\omega t + \phi)}]$

2. Fourier Transform:

Converts time-domain signals to frequency-domain: $X(f) = \int_{-\infty}^{\infty} x(t) \cdot e^{-i2\pi ft} \, dt$

The result is complex:

• Magnitude $|X(f)|$ tells you “how much” of each frequency
• Phase $\angle X(f)$ tells you “when” each frequency component peaks

3. I/Q Signals in Radio:

LoRa and other radios use I/Q (In-phase/Quadrature) representation:

• I = real part of signal
• Q = imaginary part of signal

This allows representing any modulation scheme.

15.11.3 F.3 Complex Number Operations

15.12.1 G.1 Bits and Entropy

Treat bits and entropy as measures of uncertainty, not synonyms for file size. First identify the possible outcomes and their probabilities, then calculate how much information a result conveys. The distinction connects source data, compression, cryptographic randomness, and channel capacity while keeping each claim tied to its own assumptions.

Entropy formula: $H = -\sum_i p_i \log_2(p_i)$

Where $p_i$ is the probability of each outcome.

Example - Sensor Alarm:

• Alarm triggers 1% of the time
• $H = -(0.01 \log_2 0.01 + 0.99 \log_2 0.99)$
• $H = -(0.01 \times -6.64 + 0.99 \times -0.014) = 0.08$ bits

The alarm provides only 0.08 bits of information per reading (very predictable).

15.12.2 G.2 Shannon Capacity

The maximum rate at which information can be reliably transmitted:

$C = B \log_2(1 + SNR)$

Practical Implications:

Key Insight: To double capacity, you need to roughly quadruple SNR. This explains why LoRa trades data rate for range (lower SNR = lower rate but still works).

15.13.1 Essential Formulas for IoT